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Almost everyone has some experience of how radio turns static when driving into a tunnel. The radio stations only work if a special radiating cable is installed inside the tunnel. Once upon a night, I could not sleep, pondering why the FM radios are unsuable in the tunnels. From my experience, mobile phone signals that operate at higher frequencies work better. Someone might have told me that FM radio wavelength is so large that the wave “does not fit the tunnel”. But as intuitively acceptable as this explanation could be (I don't think it is even that), this answer did not satisfy me.
After some research, I found that the explanation lies in Maxwell equation's behavior (well, what a surprise). Everything backs down to the boundary conditions of the electromagnetic field at the tunnel wall: the tangential component of the electric field component of the electromagnetic wave has to be near zero in the tunnel interface – this leads to a situation where the wave has, in a sense, no room to oscillate inside the tunnel.
Intuitive or not, this is what the equations tell us. To demonstrate this, I solved using FEM, the Helmholtz equation for a plane wave in a two-dimensional setting mimicking a tunnel and its entrance. The electromagnetic wave's polarization is so that the electrical component is pointing at the right angle w.r.t. the plane (either towards or against the reader of this page). Boundary conditions inside the 2D tunnel is zero. The coloring represents the electrical component's magnitude.
We compare the behavior of two different wavelengths:
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